The modularity conjecture for higher theta series of unitary groups

Let $X'/X$ be the fixed double cover, let $U(2m)$ be the associated quasi-split unitary group, and let $\widetilde{Z}^{r}_{m}$ be the generating series taking values in $\operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{GU(n)})$. Writing $\widetilde{H}_{m}$ for the unitary similitude group and $\widetilde{P}_{m}$ for its Siegel parabolic, the series is initially left invariant under $\widetilde{P}_{m}(F)$. Modularity conjecture. The map $\widetilde{Z}^{r}_{m}$ descends to a map

$$Z^{r}_{m}: \operatorname{Bun}_{GU(2m)}(k)\to \operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{GU(n)}),$$

i.e., the corresponding function on $\widetilde{H}_{m}(\mathbb{A})$ is left $\widetilde{H}_{m}(F)$-invariant. In other words, the Chow class $\widetilde{Z}^{r}_{m}(\mathcal{G},\mathcal{E})$ depends only on the Hermitian bundle $\mathcal{G}$ and not on its Lagrangian sub-bundle $\mathcal{E}$. For $r=0$ this follows from automorphy of the classical theta series for the dual pair $(GU(2m),GU(n))$; for positive $r$ the conjecture remains open in general, with some cases supported by modularity results for rank-one unitary groups and by the non-singular higher Siegel--Weil formula.